| L(s) = 1 | + i·5-s − 1.71i·7-s − 1.71i·11-s + 4.20·13-s + (3.91 − 1.28i)17-s + 3.33·19-s − 6.20i·23-s − 25-s − 3.71i·29-s − 3.62i·31-s + 1.71·35-s − 3.71i·37-s + 10.9i·41-s + 1.15·43-s − 9.17·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 0.646i·7-s − 0.515i·11-s + 1.16·13-s + (0.949 − 0.312i)17-s + 0.765·19-s − 1.29i·23-s − 0.200·25-s − 0.689i·29-s − 0.651i·31-s + 0.289·35-s − 0.610i·37-s + 1.71i·41-s + 0.176·43-s − 1.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.998861606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.998861606\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 17 | \( 1 + (-3.91 + 1.28i)T \) |
| good | 7 | \( 1 + 1.71iT - 7T^{2} \) |
| 11 | \( 1 + 1.71iT - 11T^{2} \) |
| 13 | \( 1 - 4.20T + 13T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 6.20iT - 23T^{2} \) |
| 29 | \( 1 + 3.71iT - 29T^{2} \) |
| 31 | \( 1 + 3.62iT - 31T^{2} \) |
| 37 | \( 1 + 3.71iT - 37T^{2} \) |
| 41 | \( 1 - 10.9iT - 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 + 9.17T + 47T^{2} \) |
| 53 | \( 1 + 7.91T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 1.79iT - 61T^{2} \) |
| 67 | \( 1 - 6.78T + 67T^{2} \) |
| 71 | \( 1 + 8.41iT - 71T^{2} \) |
| 73 | \( 1 + 6.38iT - 73T^{2} \) |
| 79 | \( 1 - 7.15iT - 79T^{2} \) |
| 83 | \( 1 + 1.83T + 83T^{2} \) |
| 89 | \( 1 + 3.62T + 89T^{2} \) |
| 97 | \( 1 - 4.84iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.925554241376838730206169988472, −7.29921194363174030617371002043, −6.31862491568440972829634867358, −6.06697849281670078738177485035, −5.01794581996673426890631286445, −4.19013168786127387472203383633, −3.40633269478720327528702930771, −2.81111864604084126220039669047, −1.50070477321755276050712810507, −0.56176115867467754822769772809,
1.17293833605922887236333773399, 1.82153566648206638954955062258, 3.15234148749533175173720939666, 3.63204252411202958112971648551, 4.70799489130361158426982311686, 5.45264880992298007939741475216, 5.87605750443466502152355512686, 6.82163405740894771858389844600, 7.57251856737053927021568291911, 8.239898402909554763568250450405
